The key to understand with Time Team is that Mick Aston is actually the lead, not Tony Robinson, despite how it may appear.
Up through and including series 18, the show has a fairly standard approach, with Mick planning the dig and Phil Harding gruffly digging things up, while Tony Robinson tries to bring together what is going on for a lay audience.
One of my favourites from series 18 is 18×02 which TVO calls Saxon Death (available until July 30, 2017), but which is actually titled Saxon Death, Saxon Gold. At the time of this writing TVO’s Time Team pages have a bit of series 17, all of 18 and 19, and series 20 is in the process of rolling out.
Starting with series 19 the show was revamped quite dramatically and made a lot more like the [Era] Farm shows, including Farm presenter Alex Langlands and the introduction of lead-in spoilers (“teaser video”) showing some of the major finds and events that are to come. There is also a reduced focus on archeologists and analysis
and a greater emphasis on re-creation, with different major contributors.
Mick Aston opposed this change, and after hanging on for series 19, in 2012 left the show he had basically originated. He passed away the year after.
Western Daily Press – Professor Mick Aston: Why I quit Time Team, and the danger of losing touch with our history – February 13, 2012
UPDATE 2016-04-11: Current Archaeology wrote an appreciation of Time Team in December 2012 that provides some history about the show and details about the series 19 changes. Time Team: the rise and fall of a television phenomenon ENDUPDATE
So if you want to watch the core of the show as originally imagined, watch series 1-18 and stop. The show ended after series 20 (March 2013) anyway. Mick was born in 1946 and Phil was born in 1950, so in any case it was pretty much time for them to retire on top of their game.
It’s a shame they couldn’t have done 20 years with Mick Aston and then wrapped up the show.
UPDATE 2016-08-16: After watching all of Time Team up to and including season 18, you can watch the Time Team Specials up to and including number 46, The Way We Lived (about how houses have changed over the centuries). After that you are into 2012 Specials, post season 18. Note that the Specials are kind of all over the map, from new content with the team, to what are basically sales pitches for major sites featuring mainly or only Tony Robinson, to clip shows.